Uniform stationary phase estimate with limited smoothness

TL;DR

This paper improves uniform estimates for oscillatory integrals with stationary phase by reducing regularity requirements and employing wave packet decomposition, which offers a geometric approach less sensitive to smoothness.

Contribution

It introduces a wave packet decomposition method to obtain uniform estimates with lower regularity assumptions on phase and amplitude functions.

Findings

Reduced regularity conditions for uniform estimates

Wave packet decomposition transforms decay estimates into geometric support disjointness

Applicable to phase and amplitude functions depending on oscillation parameters

Abstract

In this paper, we consider the uniform estimate for the oscillatory integral with stationary phase, which was previously studied by Alazard-Burq-Zuily. We significantly reduce the order of required regularity condition on the phase and amplitude functions for the uniform estimate. We also study estimates for the oscillatory integrals of which phase and amplitude functions depend on the oscillation parameter. The novelty of this article lies in the use of the wave packet decomposition, which transforms the decay estimate for the oscillatory integral to the disjointness property of the supports of wave packets. The latter is geometric in its nature and less sensitive to the smoothness of the phase and amplitude functions.